One mole of a certain ideal gas is contained under a weight-less piston of a vertical cylinder at a temperature `T`. The space over the piston opens into the atmosphere. What work has to be performed on order to increase isothermally the gas volume under the piston `n` times by slowly raising the piston ? The friction of the piston against the cylinder walls is negligibly small.

Explanation

Let A be the area of cross-section
`F+PA=P_(0)A`
`F=(P_(0)-P)A`
Work done by the agent
`W= overset(V)underset(etaV)int FdX=overset(etaV)underset(V)int(P_(0)-P)` AdX
`overset(etaV)underset(V)int(P_(0)-P)dV`

`=p_(0)(eta-1)V-overset(etaV)underset(V)intn RT(dV)/(V)`
`=RT(eta-1)-nlog_(e)eta`